In Exercises 1–40, use properties of logarithms to expand each lo...

Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.

log(1000x)

Accepted Answer

Hello. Today we're going to be simplifying the given logarithms using the properties of logarithms. So the logarithms given to us is the logarithms of million times X. Now before we continue, let's go ahead and simplify this large number. 100 million can be rewritten as 10 to the power of eight. So let's go ahead and rewrite the inside argument and change our logarithms to be the longer rhythm of 10 to the power of eight times X. Okay now we're going to need to separate this using one of our logarithmic properties. Notice how the inside argument here is using multiplication. So that's an indication that we're going to be using the property of multiplication and that states that if you have the log base A of B plus the log base A. Of C. That is going to be equal to the log base A. Of B times C. So we're gonna be using this property but going from right to left because the logarithms given to us already has a product in its argument. So we can separate it as two logarithms with the same base. But as a sum so we can separate this logo rhythm as the log base of 10 to the power of eight plus the logo rhythm of X. Now keep in mind that this is the common logarithms and whenever you have a common logarithms It automatically gets assigned a base of 10. So just keep in mind that both of these logarithms have a base of but you don't have to write it now on the left hand side, we can do one simplification here and that is to go ahead and move the exponent. So we can use the exponential rule for logarithms and that states that if you have the log base A of B to the power of C. You can bring the exponent in front of the logarithms as a product or in other words you can rewrite this as C times the log base A of B. And we're gonna go ahead and do that on the left hand portion of our some. So that's going to be eight times the logarithms of plus the logarithms of X. Now keep in mind that the that the general logarithms has a base of and there is one more simplification we can do here recall that if you have a logo rhythm which base matches its argument, you can go ahead and rewrite that as one. So the general logarithms of 10 can be rewritten as log base 10 of 10. And since the argument and the base match each other, you can rewrite this portion as one and eight times one is just going to give us eight. So to further simplify this, we are left with eight plus the logarithms of X. And this is going to be the complete simplification of our given logarithms and that means that the answer to this problem is going to be d. So I hope this video helps you in understanding how to use the properties of logarithms, and I'll go ahead and see you all in the next video.

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log(1000x)

Key Concepts

Properties of Logarithms

Logarithm of a Power

Evaluating Logarithmic Expressions

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